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Theorem mulgsubdir 15656
Description: Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgsubdir.b  |-  B  =  ( Base `  G
)
mulgsubdir.t  |-  .x.  =  (.g
`  G )
mulgsubdir.d  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
mulgsubdir  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  -  N )  .x.  X )  =  ( ( M  .x.  X
)  .-  ( N  .x.  X ) ) )

Proof of Theorem mulgsubdir
StepHypRef Expression
1 znegcl 10678 . . 3  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
2 mulgsubdir.b . . . 4  |-  B  =  ( Base `  G
)
3 mulgsubdir.t . . . 4  |-  .x.  =  (.g
`  G )
4 eqid 2441 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
52, 3, 4mulgdir 15650 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  -u N  e.  ZZ  /\  X  e.  B ) )  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) ) )
61, 5syl3anr2 1271 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) ) )
7 simpr1 994 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  M  e.  ZZ )
87zcnd 10746 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  M  e.  CC )
9 simpr2 995 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  N  e.  ZZ )
109zcnd 10746 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  N  e.  CC )
118, 10negsubd 9723 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( M  +  -u N )  =  ( M  -  N
) )
1211oveq1d 6104 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  -  N )  .x.  X
) )
13 eqid 2441 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
142, 3, 13mulgneg 15643 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( ( invg `  G ) `
 ( N  .x.  X ) ) )
15143adant3r1 1196 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( -u N  .x.  X )  =  ( ( invg `  G ) `  ( N  .x.  X ) ) )
1615oveq2d 6105 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) )  =  ( ( M  .x.  X ) ( +g  `  G ) ( ( invg `  G
) `  ( N  .x.  X ) ) ) )
172, 3mulgcl 15642 . . . . 5  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
18173adant3r2 1197 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( M  .x.  X )  e.  B
)
192, 3mulgcl 15642 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
20193adant3r1 1196 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
21 mulgsubdir.d . . . . 5  |-  .-  =  ( -g `  G )
222, 4, 13, 21grpsubval 15579 . . . 4  |-  ( ( ( M  .x.  X
)  e.  B  /\  ( N  .x.  X )  e.  B )  -> 
( ( M  .x.  X )  .-  ( N  .x.  X ) )  =  ( ( M 
.x.  X ) ( +g  `  G ) ( ( invg `  G ) `  ( N  .x.  X ) ) ) )
2318, 20, 22syl2anc 661 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X )  .-  ( N  .x.  X ) )  =  ( ( M  .x.  X ) ( +g  `  G
) ( ( invg `  G ) `
 ( N  .x.  X ) ) ) )
2416, 23eqtr4d 2476 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) )  =  ( ( M  .x.  X )  .-  ( N  .x.  X ) ) )
256, 12, 243eqtr3d 2481 1  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  -  N )  .x.  X )  =  ( ( M  .x.  X
)  .-  ( N  .x.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089    + caddc 9283    - cmin 9593   -ucneg 9594   ZZcz 10644   Basecbs 14172   +g cplusg 14236   Grpcgrp 15408   invgcminusg 15409   -gcsg 15411  .gcmg 15412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-seq 11805  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546
This theorem is referenced by:  odmod  16047  odcong  16050  gexdvds  16081  archiabllem1a  26206
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